Principal 2-blocks and Sylow 2-subgroups

Abstract

Let G be a finite group with Sylow 2-subgroup P. Navarro–Tiep–Vallejo have conjectured that the principal 2-block of N_G(P) contains exactly one irreducible Brauer character if and only if all odd-degree ordinary irreducible characters in the principal 2-block of G are fixed by a certain Galois automorphism. Recent work of Navarro–Vallejo has reduced this conjecture to a problem about finite simple groups. We show that their conjecture holds for all finite simple groups, thus establishing the conjecture for all finite groups.